In cases where such a “gap" exists, we can hope to find a substitute for graphs of curves by constructing graphs of other objects naturally associated to a surface. For instance, in the case of the plane minus a Cantor set, Bavard showed that the ray graph is connected and \(\delta \) -hyperbolic and has infinite diameter orbits under the natural action of the mapping class group [1]}. In the case of the tripod, it follows from a recent result of Fanoni–Ghaswala–McLeay that its mapping class group acts with infinite diameter orbits on its omnipresent arc graph; this is a connected, \(\delta \) -hyperbolic graph where vertices correspond to certain (possibly bi-infinite) arcs between distinct ends of the surface and where edges correspond to disjointness [2]}.
